In Fig. REF , the red curves are constantWe take the underground neutrino to be at a depth \(d=30\,\) km, however, on the log-log plot the precise value of \(d\)  used is not relevant. contours for \(|\epsilon |\) . There is a slope change near \(M_{Z^{\prime }}\simeq 10^{-13}\,\) eV where the potential energy sourced by the matter in the earth is most important and saturated, and
for \(M_{Z^{\prime }}\)  below \(10^{-17}\,{\rm eV} \sim (1\,{\rm AU})^{-1}\)  there is another jump in the red curve where the potential energy sourced by electrons in the sun becomes more important.
The Super-K experiment set an upper limit on the matter effect based on atmospheric neutrino data \(|\epsilon _{\mu \mu }-\epsilon _{\tau \tau }|<0.147\)  [1]}, [2]}, [3]} and has excluded the red shaded region. For reference, we also show the curve for \(|\epsilon |\sim 0.01\)  using the red dashed curve which perhaps may be probed by future neutrino experiments. The shaded grey region which is labelled weak gravity is where \(g^{\prime 2} <G(m_{\nu }^{\rm lightest})^2\) , when \(m_{\nu }^{\rm lightest}=0.01{\rm eV}\) .
